#149697

order	512
nilpotency class	5
isoclinism	isoclinic class of #1406
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#353
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#423	C₂³
socle	C₂	#11475
prime radical	#3150	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

330 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,11,6,10,26,37,7,13,24,39,27,12,25,40)(3,14,16,38,18,8,28,30,19,31,46,57,17,32,23,52)(4,9,20,34,21,33,50,58,22,36,48,60,51,35,49,61)(15,41,42,59,44,29,53,54,45,55,62,64,43,56,47,63);(1,3)(2,8)(4,15)(5,23)(6,18)(7,19)(9,29)(10,31)(11,38)(12,14)(13,32)(16,26)(17,27)(20,47)(21,44)(22,45)(24,28)(25,46)(30,37)(33,55)(34,59)(35,41)(36,56)(39,57)(40,52)(42,50)(43,51)(48,53)(49,62)(54,58)(60,64)(61,63);(1,4,6,21,7,22,27,51)(2,9,10,33,13,36,12,35)(3,15,18,44,19,45,17,43)(5,20,26,50,24,48,25,49)(8,29,31,55,32,56,14,41)(11,34,37,58,39,60,40,61)(16,42,28,53,46,62,23,47)(30,54,57,64,52,63,38,59)

Polycyclic presentation:

2^9|a^2=e,c^2=g,d^2=h,e^2=g,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=dh,e^b=efhi,f^b=fi,e^d=ehi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56893);